Definitions of cloud computing vary, but it basically boils down to using computing resources that sit on the public network. That doesn't mean everything has to be out on the "cloud" with your PC acting as a dumb terminal.
For example, you might have a spreadsheet that manipulates stock data according to your own calculations, but retrieves the raw data directly from the 'net (without you having to download and convert it).
If you're a gamer, steamworks is effectively a cloud. Your PC runs the game, but gamesaves are stored on Valve's computers.